Monday, 13 February 2012

After Stephen Law’s debate with William Lane Craig there was a lot of discussion on the web about the efficacy of Law’s argument, the Evidential Problem of Evil, in countering Craig’s arguments.

Many of those who think the Evidential Problem of Evil ineffective argue that it is a probabilistic argument that, at best, shows that God is unlikely. (e.g. Chab123 2011) Others (e.g. Stuart 2011) felt that insufficient attempt was made to show what was wrong with Craig’s arguments for God, especially the Kalam Cosmological Argument.

Craig’s opposing arguments for God are deductive arguments. If the premises are true and the logic valid the conclusion is conclusively established: it is simply impossible for the conclusion to be false.

The best a “probabilistic” argument can do is make it very, very unlikely that the conclusion is false. And, as Sherlock Holmes pointed out to Dr. Watson, no matter how unlikely a conclusion seems if nothing else is possible then it must be true.

This is Chab123’s complaint. Law claimed that the Evidential Argument from Evil, when plugged into the Modal Ontological Argument, demonstrated the impossibility of God:

“His confusion on this point is breath-taking. His evidential argument from evil, at its very best, shows, at most, that it is probable that God does not exist. The probability is less than 1. To defeat the ontological argument with an argument from evil, his argument would have to entail that God does not exist. The probability that God does not exist would have to be 1. It would have to prove, as he says, that the conclusion of Craig’s argument is false. But Law’s own argument, as a matter of logic alone, cannot achieve this goal. It is a probabilistic argument. As such, it leaves open the possibility that God exists, even if the probability is quite low.”

Nothing probabilistic about modus tolens

Craig’s arguments make use of a rule of logic called modus ponens. The rule is:

If x then y

x

Thus y

The picture many people have of a “probabilistic” argument is along the lines of:

The first swan seen is white

The second swan seen is white

The third swan seen is white

…

The nth swan seen is white

Thus all swans are white

It can seem that the Evidential Problem of Evil follows this pattern. Just as the first white swan is not enough to establish that all swans are white the first bit of gratuitous suffering is not enough to cast doubt on God. The seemingly endless succession of uniformly white swans seems to make it likely that all swans are white and the endless widespread pointless suffering seems to make it likely that there is no God.

In fact not all swans are white, despite how overwhelmingly improbable that seemed before the discovery of Australia. “y” in the example above, cannot be wrong, “all swans are white” both can be, and is.

The Evidential Argument from Evil, though, is not a “probabilistic” argument. It is just as deductive as Craig’s arguments; making use of a rule of logic called modus tolens.

Modus tolens runs:

If x then y

Not y

Thus not x

And the Evidential Problem of Evil runs like this:

If God existed then there wouldn’t be so much pointless suffering in the world

There is so much suffering in the world

Thus God does not exist.

This is just as “logically airtight” as any of Craig’s arguments; if the premises are accepted then the conclusion must be accepted. We can, of course, dispute the premises (I am, though, persuaded by Law’s “evil god hypothesis” (Law 2009) that rejection of them would be irrational). As for the comparison arguments from Craig: though in the above example of modus ponens the conclusion was absolutely certain, the premises were just assumed to be true and the logic was kept simple to ensure its validity. Craig's premises, and the validity of his logic, are both at risk: they are not at all certain and that uncertainty carries through to the conclusion.

The idea that the Evidential Argument from Evil is a member of some poor-relation-species of argument compared to Craig’s is erroneous.

But you haven’t refuted Craig’s arguments!

The complaint that Law did not adequately argue against the Kalam Cosmological Argument is equally erroneous.

When criticising an argument we could “undermine” it: dispute the premises or find fault with the logic. Or we could show it to be false. Law certainly did not explain why the Kalam Cosmological Argument concluded falsely but he did give a good argument that it does conclude incorrectly.

If the Kalam Cosmological Argument is a sound deductive argument then no evidence will count against its conclusion. That evidence does count against its conclusion is good evidence that it’s not a sound deductive argument!

Similarly, with the Modal Ontological Argument. This, very briefly, assumes the possibility of God and concludes with His existence. If one accepts the logic of the argument (I don’t) then, far from showing “breath-taking” confusion Law’s claim is perfectly cogent:

If God is possible then God exists (Modal Ontological Argument)

God does not exist (Evidential Argument from Evil)

Thus God is not possible

Of course we would like to know where the argument falls down, knowing where the argument falls down may expand our knowledge and that knowledge may help us avoid error in future arguments. But look at this, readily criticised, argument:

All elephants are pink

Socrates is an elephant

Socrates is mortal

The “logic” is hardly worthy of the name, the premises are junk but the conclusion is true. A demonstration of the falsity of the premises or the invalidity of the logic does not establish that the conclusion is false; a demonstration of the falsity of the conclusion, obviously, does.

The Evidential Argument from Evil is a better counter argument than picking holes in Craig's logic and questioning his premises. Whilst direct criticisms of Craig's arguments may cast doubt on those arguments the Evidential Problem of Evil shows them to be false.

William Lane Craig seeks to put forward deductive arguments for the existence of God.

If Craig gets his logic right he will have produced a valid deductive argument. If an argument is valid then if the premises are true the conclusion is true. Validity is not enough though, Pooh's argument here is valid:

"And if anyone knows anything about anything," said Bear to himself "it's Owl who knows something about something," he said, "or my name's not Winnie the Pooh," he said. "Which it is," he added. "So there you are." (Milne, 1977)

We can only be sure of the truth of the conclusion if the argument is valid and we are sure of the truth of the premises. Now Craig
cannot, none of us can, establish the truth of his premises beyond doubt. What he seeks to do is adopt premises that
are “more plausible than their negations” (2010)

This is
problematical. Accepting that the
premise adopted is more plausible than
its negation may lead the reader, or debate attendee, to believe that the
premise adopted is the most plausible
premise. Were the only options to be that
premise and its negation this would be the case. Take, however the Kalam Cosmological
Argument:

1. Everything that begins to
exist has a cause.2. The universe began to exist.3. Therefore, the universe has a
cause(Craig
2010)

Other premises

There are a number of alternative first premises other than "not everything that begins to exist has a cause", each with good arguments for its adoption:

Alternate A: Not everything has a cause

Argument for: We might consider that "everything has a cause". However causes happen before effects, meaning that each effect has a predecessor. As each cause is "something" and requires it's own cause, a cause which also requires a cause further back. The regress, if everything had a cause, would never terminate. There are good arguments for this being impossible: meaning that as there is something not everything had a cause. (Note, for later, that if an effect can precede a cause or be simultaneous with its cause there is no obstacle to everything having a cause")

Alternate B: Everything we can understand has a cause

Argument for: Though we speculate about entities we don't understand we shouldn't introduce that speculation into a deductive argument as, well, we don't understand them. We can confidently say that anything comprehensible has a cause as causality is part of the way that we understand things.

Alternate C: For any X : if a Plank second of time occurred before X's existence then X had a cause.

Argument for: Again, causes happen before effects. It appears that a Plank second is the shortest period of time any cause can be before its effect. For any effect, therefore, there has to be at least one Plank second before it for its cause to be in.

There we have three alternatives to the first premise given by Craig.

Whilst it is difficult (probably impossible) to quantify plausibility, each of the alternatives appears to be at least as plausible as the premise given by Craig. It would be difficult to argue that together they are not considerably more plausible than the premise given by Craig. To argue that they were not more plausible would require arguing that:

"Alternate A or Alternate B or Alternate C" is more plausible than "Craig's first premise"

Now "Alternate A or Alternate B or Alternate C" fails to establish the conclusion "the universe had a cause", whilst Craig's first premise does. But "Alternate A or Alternate B or Alternate C" is more plausible than Craig's first premise. Limit our choice to these alternatives and the following is true:

The most plausible conclusion is that Craig's conclusion has not been established

What Craig is arguing for.

Supposing we come to the debate holding Alternate C. What must Craig show in order to establish his contention that the universe had a cause? Some of our position (Alternate C) and Craig's overlap, its just that that part of Craig's position fails to establish his conclusion. What Craig needs to convince us of is the difference between his first premise and ours. What is that difference? We already agree that all stars, all planets, all living things, all atoms and sub atomic particles, all light and, if they exist, all dark matter and dark energy have causes. We accept that everything arising since the first Plank second of the universe's existence had a cause.

Our dispute is about the first Plank second of the universe.

And what is the first Plank second of the universe's existence? Is it anything more than the beginning of the universe? Is there anything else to attribute a cause to other than "the universe"? In order to persuade us to accept his premise over ours does Craig not have to persuade us that the universe had a cause? In other words, to establish that the universe has a cause he first has to establish that the universe has a cause in order to establish his premise that everything that began to exist had a cause.

And what does he have to do to establish this? He has to persuade us that the universe had a cause even though there was no time for the cause to be in. If he succeeds in doing that he must either allow simultaneous causes (which allows us to curtail the infinite regress mentioned in Alternate A, which allows "everything has a cause") or exempt the universe from this rule. If we can exempt the universe from the rule to have a prior cause we can just as well exempt the universe from having a cause.

Conclusion

It is the dichotomy created by the "more plausible than its negation" that allows other, more plausible, alternatives to be ignored and Craig's need to argue for a cause of the universe tout court to remain hidden. It's the dichotomy which allows a startlingly implausible argument to look plausible.

Friday, 10 February 2012

In which I find that the Modal Ontological Argument "works" because Modal Logic languages cannot cope with modalities about modalities. I fix that.

Like its non-modal counterpart,
Anslem’s plain old “Ontological Argument”, the Modal Onotological Argument can
be maddening. Whilst both seem obviously
fallacious it is not at all obvious where the fallacy lies. Is it a parlour trick? If so, where is the sleight?

The argument seems to commence with the assumption that God possibly
exists. Then with no further assumptions
than conventional modal logic, the argument concludes that God actually exists. The assumption that God possibly exists seems
innocuous enough, the claim that God actually exists is a little more
contentious. As the only bridge between
the two is modal logic it suggests that the fault lies there.

That there is a fault can be shown by running the argument for other entities
we are quite sure do not exist but will accept that they could possibly exist;
such as The Golden Mountain, the Flying
Spaghetti Monster or the Decent Pint of Mass Produced Lager.

A version by William Lane Craig,
itself a slightly reworded version of Alvin Plantinga’s “victorious” argument,
goes like this:

1. It is possible that a
maximally great being exists.

2. If it is possible that a maximally
great being exists, then a maximally great being exists in some possible world.

3. If a maximally great being exists
in some possible world, then it exists in every possible world.

4. If a maximally great being exists
in every possible world, then it exists in the actual world.

5. If a maximally great being exists
in the actual world, then a maximally great being exists.

6. Therefore, a maximally great being
exists.

(Craig, 2010) (Plantinga, 1974, p.214-217)

Possible Worlds

Modal logic is expressed both
formally in symbolic language and less formally in everyday language. In everyday language the modal statements are
expressed by means of statements about “possible worlds”, complete and discrete
descriptions of the way the world could be.
If p is possible there will be
a complete and discrete description of the way the world could be that contains
p: p will “exist in” a “possible world”.

If p is necessary then every description of the way the world could be
will include p: p will exist in all possible worlds. “Impossible” can be expressed by absence from
all possible worlds. “Actual” involves
existence in one, special, possible world: the actual world. (As the actual
world is such a special world it deserves its own name and we shall follow Plantinga
in calling it “Kronos”).

“It is possible that a
maximally great being exists”

A “maximally great being” takes
the place of God in the argument.
“Maximal greatness” is “maximal excellence” in every possible
world. “Maximal excellence” is the
collection of those God-like properties of omniscience, omnipotence and moral
perfection (“OOM”).

The “being” is mentioned partly
because this is an argument about a being, God, but mostly to avoid issues of
trans-world identity. OOM are attributes
of beings or nothing at all. If OOM
exists in a possible world then a being exists in that world to possess
them. The existence of OOM entails a
being but the existence of OOM in more than one world does not entail the same
being in each one. The disadvantage in
talking about OOM rather than a specific OOM being is that you might prove that
a god is in each possible world without
showing that God is in each possible world.
The advantage of discussing OOM alone is that it makes the discussion
that follows a lot simpler. So long as
we undertake not to introduce a trans-world-identity sleight, no one should
mind us simplifying matters by talking directly of OOM.

There is a claim of OOM qualified
by the modality, necessity, conferred by OOM being subsumed in “maximally
greatness”. There is also a
qualification by possibility.

What does the “possible” qualify,
what is its range? The range of the
necessity is clear: OOM. But does
“possible” range over OOM or over “necessary OOM”?

If “possible” ranges over OOM
there is a claim that OOM is possible.
There is also, clearly, a claim of necessary OOM. Two claims about OOM are made and the modal
claims of the first premise can be re-phrased:

1a. OOM is
possible and necessary

Expressing this in terms of
possible worlds is simple: OOM is in all of them.

The alternate reading is that the
“possible” qualifies the necessity of
OOM (call that “OOM*”):

1b.
OOM* is possible

It is much more difficult to
express just what the modal claim is here.
The modality of OOM* is not the modality of OOM; a claim that OOM* is
possible is not a claim that OOM is possible.
For example, let p be a
plainly contingent entity; such as the piece of paper or the screen you are
reading this on. Let p* be the claim that p is necessary. p* is plainly impossible, p
is a contingent entity and contingent entities are not necessary. p, though,
is plainly possible: if it exists, it exists in Kronos, which is a possible
world, so it's possible.

Extending the language

Expressing these modalities of
modalities within the language of possible worlds sketched out above is more
than tricky, it may require an extension of that language. The possible worlds we are used to for
speaking of modalities are populated by actuals. A possible p is actually in a possible world.
To talk about the modality not of an actual but of a modality, to talk meta-modally,
somewhere populated by modalities is required.

Modalities populate the set of possible worlds.

Now the language of possible worlds
requires, in a way, complete description of all possible worlds. We need the full set of possible worlds to
“look into” to check whether p is
present in all, some or none. So, when
asked whether p is possible we rustle
up the full set of possible worlds, “S1”,
and have a look.

But S1 may be wrong, it may not be the set of possible worlds, just a set of worlds that could be the
set of possible worlds. We have imagined
a possible set of possible worlds, a set in which the modality of p resides but only possibly.

The full set of sets of possible
worlds is where modalities of modalities reside.

- To say that X is possibly [a particular modality] is
to say that in at least one set of possible worlds X has that modality.

- To say that X is necessarily [a particular modality]
is to say that in all sets of possible worlds X has that modality

- To say that X is impossibly [a particular modality]
is to say that X has that modality in
no sets of possible worlds

- To say that X is actually [a particular modality] is
to say that X has that modality in the
actual set of possible worlds (meta-Kronos).

Extending symbolic modal systems

To express this formally we need
to distinguish modalities and meta-modalities.
We shall add subscripts to the modal operators, □ (necessity) and ◊
(possibility). An operator that is not subscripted is taken to be subscripted
with zero.

We need to make clear the range of the meta-modalities. This we shall do by restricting the permissible
expressions in the extended language, "well formed formulae" or
"wff". We shall introduce two
rules:

1. A wff in whichever “standard”
modal logic we are extending is a wff.

2. A wff enclosed in brackets
with a modal operator to the left with a subscript one higher than the highest
subscripted operator within the brackets is a wff.

Finally we need to be able to work on a modal statement unencumbered
by the meta-modality, to figure out what we are qualifying before applying the
meta-logic. We shall add a new
argumentation rule, the "detachment/re-attachment rule":

- On a line in an argument the
operators with the highest subscript may be removed ("detachment")

- If the operators are detached
they must be reintroduced on a later line to form a wff
("attachement")

- When attached the operators are
given the lowest subscript then available.

The confusion in the argument

We can now clearly distinguish three different readings of
the first premise. We can express them
both formally and in the more everyday language of possible worlds. Our opening
position clear we can proceed to argue modally.

The first reading, 1a: OOM is possible and necessary, is
expressible in the non-extended language.
OOM is in all possible worlds and every possible world. As it is in all possible worlds, it is in
Kronos. Therefore OOM exists.

Formally:

1.◊OOM
& □OOM (Premise)

2.□x
→ x (Axiom)

3.□OOM
(1 Elimination)

4.OOM
(2,3)

The second reading, 1b: possibly OOM*, requires the
extensions to the languages. OOM*, the
necessity of OOM, entails the presence of OOM in each world in a set of
possible worlds. In that set of possible
worlds OOM will be in Kronos and will, therefore, exist. OOM is actually exisiting in at least one
possible set of possible worlds and is, thus, possibly existing.

Formally:

1.◊1(□OOM)
(Premise)

2.□x
→ x (Axiom)

3.□OOM
(Detachment)

4.OOM
(2,3)

5.◊OOM
(Attachment)

A third reading, not mentioned above, is the meaning someone
probably “hears” on first encountering the argument:

1c
God is possible

In the language of possible worlds, God is in at least one possible
world. So God is possible.

Formally:

1.◊G
(Premise and conclusion)

Notice how 1b reaches the same conclusion as the premise
that is “heard” and which will be readily agreed. The unextended languages, though, are unable to
adequately express 1b. As a result any
formalisation, “full” or “everyday language”, is
forced onto 1a, or similar. 1a naturally
reaches the conclusion of the Modal Ontological Argument but, if made clear
would not be assented to by atheists, agnostics and, even, many theists.